$\textbf{Match the following columns} \hspace{1cm}$ ($R$ = radius, $k$ = radius of gyration) $…

$\textbf{Match the following columns} \hspace{1cm}$ ($R$ = radius, $k$ = radius of gyration) $ \begin{array}{|c|l|c|l|} \hline \textbf{Column I} & \textbf{Description} & \textbf{Column II} & \textbf{Value of } k \\ \hline \text{(A)} & k \text{ for a solid sphere rotating about its tangent} & \text{(P)} & \sqrt{2} R \\ \hline \text{(B)} & k \text{ for a ring rotating about its tangent perpendicular to its plane} & \text{(Q)} & \dfrac{R}{2} \\ \hline \text{(C)} & \begin{array}{l} k \text{ for a uniform solid right circular cone} \\ \text{rotating about its central axis} \end{array} & \text{(R)} & \dfrac{\sqrt{7}}{\sqrt{5}} R \\ \hline \text{(D)} & k \text{ for a uniform disc rotating about its diameter} & \text{(S)} & \dfrac{\sqrt{3}}{\sqrt{10}} R \\ \hline \end{array} $
  1. $(\mathrm{A})-(\mathrm{p}) ;(\mathrm{B})-(\mathrm{R}) ;(\mathrm{C})-(\mathrm{Q}) ;(\mathrm{D})-(\mathrm{s})$
  2. $(\mathrm{A})-(\mathrm{p}) ;(\mathrm{B})-(\mathrm{Q}) ;(\mathrm{C})-(\mathrm{s}) ;(\mathrm{D})-(\mathrm{P})$
  3. $(\mathrm{A})-(\mathrm{Q}) ;(\mathrm{B})-(\mathrm{R}) ;(\mathrm{C})-(\mathrm{p}) ;(\mathrm{D})-(\mathrm{S})$
  4. $(\mathrm{A})-(\mathrm{R}) ;(\mathrm{B})-(\mathrm{P}) ;(\mathrm{C})-(\mathrm{S}) ;(\mathrm{D})-(\mathrm{Q})$

Solution

$\textbf{Correct Matches:} $ $ \begin{aligned} \text{(A)} &\rightarrow \text{(P)} \quad (\text{Solid sphere: } I = \dfrac{2}{5}MR^2 + MR^2 = \dfrac{7}{5}MR^2 \Rightarrow k = \sqrt{\dfrac{7}{5}}R = \dfrac{\sqrt{7}}{\sqrt{5}} R) \\ \text{(B)} &\rightarrow \text{(Q)} \quad (\text{Ring about tangent: } I = 2MR^2 \Rightarrow k = \sqrt{2} R) \\ \text{(C)} &\rightarrow \text{(R)} \quad (\text{Cone about axis: } I = \dfrac{3}{10}MR^2 \Rightarrow k = \sqrt{\dfrac{3}{10}}R = \dfrac{\sqrt{3}}{\sqrt{10}} R) \\ \text{(D)} &\rightarrow \text{(S)} \quad (\text{Disc about diameter: } I = \dfrac{1}{4}MR^2 \Rightarrow k = \dfrac{R}{2}) \\ \end{aligned} $

Asked in: MHT CET 2022 (07 Aug Shift 2)

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